Fermat's famous theorum: proved at last?

A year ago, mathematician Andrew Wiles of Princeton University faced a troubling gap in the logic he had followed to prove Fermat's last theorem. Now, he has apparently found a way to bridge the gap and complete his proof of Fermat's famous conjecture.

Last week, in a move that caught the mathematical community by surprise, Wiles began distributing copies of two new manuscripts addressing the concerns that had been raised about his original argument. The first, lengthy paper announces the revised proof, still following quite closely the strategy Wiles had outlined in his lectures in June 1993 at the University of Cambridge in England (SN: 7/3/93, p.5).

The second, short paper, produced in collaboration with Cambridge mathematician Richard L. Taylor, contains mathematical reasoning justifying a key step in the main proof. Instead of solving the original problem, Wiles and Taylor avoided it by using a different approach to reach the same conclusion.

Both papers have been submitted for publication in the ANNALS OF MATHEMATICS.

"The proof looks really beautiful, but it's too soon to comment in detail [on its validity]," says Fernando Q. Gouvea of Colby College in Waterville, Maine. "Everybody's being very cautious." Gouvea has been attending a seminar at Harvard University, where Taylor has been discussing aspects of the proof.

Pierre de Fermat's claim, made more than 350 years ago, was that for each whole number greater than 2, the equation [x.sup.n] + [y.sup.n] = [z.sup.n] has no solutions that are positive whole numbers. Over the centuries that followed, many mathematicians tried to prove Fermat's conjecture but invariably failed.

The attack chosen by Wiles relied on recently discovered links between Fermat's conjecture and the theory of elliptic curves. By assuming that Fermat's last theorem is false, mathematicians could construct a "weird" elliptic curve that they believe, for other mathematical reasons, shouldn't exist.

Moreover, the existence of this strange curve would also contradict the Taniyama-Shimura conjecture, which involves other characteristics of elliptics. Hence, if proving certain aspects of the Taniyama-Shimura conjecture excluded the strange curve, this would establish the validity of Fermat's last theorem.

This is the course that Wiles followed. But his original argument foundered near the end, where he had relied on a powerful new method developed by Viktor A. Kolyvagin of the V.A. Steklov Institute of Mathematics in Russia to serve as a kind of mathematical bookkeeping system. Wiles found a serious flaw in his attempt to construct the so-called Euler system necessary for completing his proof (SN: 12/18&25/93, p.406; 6/25/94, p.406).

To cirumvent the problem, Wiles decided to go back to an approach that he had tried several years earlier but abandoned in favor of Euler systems. Helped by Taylor, who had come to Princeton last spring to work on the problem, Wiles succeeded in establishing the conditions he needed to complete the proof. This time, he used mathematical techniques based on Hecke algebras.

"The new approach turns out to be significantly simpler and shorter than the original one," noted Karl Rubin, who is presently visiting Harvard, in a message last week to colleagues at Ohio State University in Columbus. Some mathematicians are already looking into simplifying the proof further, even as they check the work done by Wiles and Taylor.

"While it is wise to be cautious for a little while longer, there is certainly reason for optimism," Rubin adds.

As the manuscripts go into circulation in the mathematical community, many mathematicians are getting a chance to see for themselves.